Theorem REF  asserts that the foliation of stable/unstable manifolds \(W_{loc}^{s/u}(S)\)  by lower dimensional stable/unstable manifolds \(w^{s/u}_{loc}(z)\)  associated to points \(z \in S\) , also perturb in a regular fashion. The local positive/negative invariance of the families \(\lbrace w_{loc}^{s/u}(z_\varepsilon )\rbrace _{z_\varepsilon \in S_\varepsilon }\)  is described in assertions (i)-(ii), and sketched for the stable foliation in Figure REF . Statements (iii)-(iv) assert the exponential contraction and repulsion along stable and unstable fibers respectively. This explains the common use of terms like “stable", “unstable" and “normally hyperbolic" when describing perturbed fibers or manifolds like \(w_{loc}^{s/u}(z_\varepsilon )\) , \(W^{s/u}_{loc}(S_\varepsilon )\)  and \(S_\varepsilon \) . Notice by assertions (v)-(vi) that stable/unstable fibers \(w_{loc}^{s/u}(z_\varepsilon )\)  are only \(C^{r-1}-\) smooth with respect to variation of the base point \(z_\varepsilon \in S_\varepsilon \) , even though by Theorem REF  the stable/unstable manifolds \(W^{s/u}_{loc}(S_\varepsilon )\)  themselves are \(C^r-\) smooth. This is also true of foliations with Fenichel slow manifolds as base in the ODE setting [1]}, [2]}, [3]}, [4]}. In contrast to the ODE setting, however, an additional nondegeneracy condition is required in assertion (v) in order to infer \(C^{r-1}-\) smoothness of the stable fibers in particular.
